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An Elegant Lambert
Algorithm
for Multiple Revolution
Orbits
by
Laura A. Loechler
S.B. Massachusetts
Institute
of Technology (1984)
Submitted
to the Department
of
Aeronautics and Astronautics in
Partial Fulfillment
of the
Requirements for the Degree of
Master of Science
in Aeronautics and Astronautics
at the
Massachusetts
Institute
of
Technology
May 1988
© Laura A. Loechler 1988
The
author hereby
grants to M.I.T.
permission
copies of this thesis
document
in
Signature
of Author
Certified
by _____
to reproduce
and to distribute
whole
or in
part.
Department
of Aeronautics
and Astronautics
May 18, 1988
Dr.
Richard H. Battii, Adjunct
Professor
,r , / TWozic corl'lvP-;SOr
Accepted
by _____________
ei:,.---'- - rroressor t-arola Y. wactlman
Chairman, Departmental
Graduate
Committee
M USEMTS
INSTATE
Of
TCLY
A -
JUN 011988
i"i
t
~~~~~~~
OPAM
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An Elegant Lambert
Algorithm for Multiple
Revolution
Orbits
by
Laura A. Loechler
Submitted
to the Department of Aeronautics and Astronautics
on May 18,
1988
in partial fulfillment of the
requirements
for the Degree
of Master of
Science in Aeronautics
and Astronautics.
Abstract
A simple efficient algorithm to solve the non-unique multiple
revolution time-constrained
two-body two-point orbital
boundary value
Lambert's problem is
of great interest for
spacecraft interception
and
rendezvous
and interplanetary
transfer. The multiple revolution
transfer
orbits permit greater
navigational flexibility and result in lower
energy orbits
than the unique direct
transfer orbit. Two
algorithms, based
on the elegant
algorithm for direct
orbital transfer developed by R. H. Battin
from Gauss'
orbit determination
method, definitively
separate the low and
high energy
solutions for each
complete revolution within the transfer. Both
retain the
excellent convergence
characteristics of the
direct transfer method,
requiring
only one additional
iteration to converge
to an additional
four significant
figures
and having nearly uniform convergence
over a wide
range of
problems.
First
the derivation of
the direct transfer algorithm is summarized.
A
simple approximation
of the maximum number
of complete
revolutions
that
can
be made within
the given transfer time
is suggested.
Both the high and
low energy algorithms
are derived
along
the lines of the direct transfer
algorithm. Tables
display the
number of
iterations required for convergence
for a large number
of transfer problems, demonstrating the capability
of these
algorithms.
Thesis Supervisor:
Dr. Richard H. Battin
Title: Adjunct
Professor of Aeronautics and Astronautics
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Acknowledgements
Thanks to my advisor,
Prof. Battin,
for his wonderful improvements
on Gauss' method which made this thesis possible and his help and
suggestions
in pursuing
various, fruitful and fruitless,
lines
of thought.
Special thanks
to Brent Lander for his help with my
Macintosh and his
LaserWriter.
The concluding year of my
Master's studies has
been sponsored by my
employer,
the Naval Research
Laboratory
of Washington, D.C., through
its
Select Graduate
Study Program.
Thanks to my supervisor
Tom Lawton
and
coworker Jim
Young for their support
and efforts in
helping me obtain
funding and time
for my education.
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Table of Contents
Chapter
1
Chapter 2
Chapter
3
Chapter 4
Chapter
5
Chapter 6
Chapter
7
Abstract
..............................................................................................
2
Acknow ledgem
ents .............................................................................
3
Table of Contents
.................................................................................
4
List of Figures
.. .............................................................................
List of Tables .......................................... . 6
Introduction
..........
.......................................................
7
Direct
Transfer
Algorithm ..................................................
9
2.1 Lambert's Problem Definitions .................................................
2.2 Geometrical Transformation of
Orbit ......................................
0
2.3
Transformation
of Time-of-Flight
to Cubic
Equation ..........12
2.4
Use of Free Parameter
to
Flatten
Cubic Equation ..................
16
2.5 Solution
of Cubic
Equation
.............................................
18
2.6 Calculation
of Orbital Parameters .............................................
9
2.7 Convergence
Behavior
.............................................
21
Time-of-Flight
Equation
with Multiple
Revolution
Orbits ......23
3.1
Addition
of Multiple Periods
..............................................
23
3.2
Estimation of Maximum
Number
of Revolutions ...............24
3.3
Relative
Energy of the Two
Solutions
per
Revolution ........28
Algorithm
to Determine
Low Energy Solution
.......................
29
4.1
Revised
Cubic Equation...............................................................29
4.2 Choice
of
Free Parameter
to Flatten
Cubic
Equation.............
30
Algorithm
to Determine
High
Energy
Solution
.........
33.............33
5.1 Redefinition
of y and Revised
Cubic
Equation
......................33
5.2
Choice of
Free Parameter
to Flatten
Cubic Equation
.............
34
5.3 Solution
of Cubic
Equation
.
...................................
36
Behavior
of Multiple
Revolution Algorithms
.............................
8
6.1 Determination of Low and High Initial Estimates ................38
6.2
Summary of Algorithms
..............................................
40
6.3 Convergence
Characteristics
..............................................
41
Conclusion............................................. 46
References............................................. 48
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List
of Figures
Figure
1 Geometry
of Elliptic Lambert Problem
............................................
Figure 2 Transformed Elliptical Orbit ..............................................................1
Figure 3
Time-of-Flight as a Function
of x for
Direct Transfer
..................14
Figure
4 Solution
Path for Direct Transfer
............................................
15
Figure
5 Solution
Path for Direct Transfer
Using Free Parameter
............
17
Figure 6 Time-of-Flight
as a Function
of x for Multiple Revolution
Orbits
.......................................................................................................
24
Figure 7 xm
and Approximation
as a Function of X.....................................26
Figure 8
Figure 9
Figure
10
Figure
11
Figure
12
Figure 13
Figure 14
T
m
and Approximations
as
a Function
of ...................................
7
Semimajor
Axis
as a Function
of x .......................................
28
Solution Path for Multiple
Revolutions
........................................
9
Solution
Path for Multiple
Revolutions
Using Free
Parameter ............................................
................................
31
Solution Path for
Multiple Revolutions
Using Redefined
y ............................................................................................................... 33
Solution
Path
for Multiple
Revolutions
Using
Redefined y and Free Parameter
. ...........................
3.......36
Composite
Low and High Energy Solution
Paths .......................
38
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List of Tables
Table1
Number
of Iterations
to Compute
Direct
Transfer
Solution
to
Eight/Twelve Significant
Figures
..............................
21
Table 2 Number
of Iterations to
Compute
Low
Energy Solution
for N = 1 to
Eight/Twelve
Significant
Figures
..............................
1
Table 3
Number
of Iterations
to Compute
Low
Energy Solution
for N
= 2 to Eight/Twelve
Significant
Figures
..............................
2
Table 4
Number
of
Iterations to Compute
High
Energy Solution
for N = 1
to Eight/Twelve
Significant
Figures
..............................
3
Table 5
Number
of
Iterations to Compute
High Energy
Solution
for N =
2 to Eight/Twelve
Significant
Figures
..............................
4
Table
6 Number
of Iterations
to Compute Solutions
for
N =
1 at
101% of Tm to
Eight/Twelve
Significant Figures
..........................
45
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Chapter 1 Introduction
The time-constrained
two-body
two-point orbital boundary value
problem, known as Lambert's problem,
has historically
been of interest in
determining the elements of
the orbits of planets and comets. Johann
Heinrich Lambert
(1728-1779)
discovered that the time to transverse an
elliptic
arc, actually any conic,
depends on only the semimajor axis and
the
distances from the initial and
final points to the
center of force and to
each
other. Leonhard Euler
(1707-1783)developed
the theorem for the
parabolic
case
and Joseph-Louis Lagrange
(1736-1813) proved the theorem
for elliptic
orbits. It is now
of practical interest in spacecraft interception
and rendezvous
and
interplanetary
transfer.
Carl Friedrich Gauss
(1777-1855)was interested in determination
of the
orbit of the asteroid Ceres from observations taken over a few
days. He
developed an elegant
computational method for orbits with a small
transfer
angle between terminal points which is singular for a 1800 transfer angle.
The limitations of Gauss' method make
it impractical for a wide range
of modern navigation problems.
Richard H. Battin made a geometric
transformation of the orbit that removed the singularity for the 180
°
transfers
and with Robin M. Vaughan developed an algorithm similar to Gauss' that
has
good convergence for 0
°
to 3600 transfers. Like Gauss, Battin makes no
assumptions regarding the type of orbit: method works elegantly for
hyperbolic,
parabolic, and elliptic orbits as constrained
by geometry
and time-
of-flight.
We
are interested in building on this algorithm,
hereafter called the
Direct Transfer
Algorithm, to exploit
problems in which the
time-of-flight is
long enough that there
exist elliptic orbits
for which the time-of-flight is
greater than one period. Then the orbit determination problem is not unique.
In general
each complete
revolution adds two
solutions. Utilizing these
multiple revolution transfer orbits provides
some navigational flexibility and
results
in lower energy
transfer orbits.
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The two solutions
for each possible complete revolution
can easily be
determined
from two algorithms
which
are simple extensions
of the direct
transfer
algorithm,
using the same geometric
transformation
and
a successive
substitution iterative
method,
and
which have the same
or better
convergence than the direct
transfer algorithm.
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Direct
Transfer
Algorithm
2.1
Lambert's
Problem
Definitions
Because the multiple
revolution
algorithms
are extensions
of
the direct
transfer
algorithm,
consider
a
brief derivation
of
the direct
transfer
algorithm.
Figure
1
Geometry
of
Elliptic
Lambert
Problem
For simplicity
consider only
elliptic
orbits as shown
if
Figure
1, although
the
direct
transfer
algorithm
is valid
for
all orbits.
Define:
At
time-of-flight
_2}
position
vectors
of the initial
point
P
1
and the
terminal
point
P
2
from
the center
of force
F
rl= Irll
r
2
=
Ir
2
1
c = Ir2- I
a
distance
from
F to P
1
and
P
2
respectively
distance from P
1
to
P
2
semimajor
axis
of orbit
9
Chapter
2
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am
smallest possible
semimajor axis of
any orbit
connecting
P
1
and P
2
[(r + 2 + c)]
s = {(
2
semiperimeter of
the triangle
FP
1
P
2
2a
m
0
angle from
r1 to
2
For
convenience
define the non-dimensional
parameter X
(-1
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Figure
2 Transformed
Elliptical Orbit
The equation for the pericenter radius must now equal that of
the
mean point r
0
,
1
r = a (1 -e
0
) = rop sec2 4 (E
2
- E
l
)
(2.1)
where eo is the eccentricity of the transformed
orbit and rop is the mean point
of
the parabolic orbit connecting P
1
and P
2
,
2
1 2
rop =
4
(rl + r 2 cos =4
s
(1 + )2 (2.2)
The true anomaly of the transformed orbit is a function of constants of
the transfer,
3
1 0
2'c (1 - 2)
_ _cos
2 %,
sinf = 1 (12) cosf = (r r
2
) -(1+%2) (2.3)
2- ( 1r+ 2) (r +r2) + 2)
libid., eqn. 6.79
2
ibid., eqn. 6.77.
3
ibid., eqn. 7.79.
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Since the eccentric anomaly of the mean point is the arithmetic mean of E
1
and E
2
, the eccentric anomaly E corresponding to f must clearly be half the
difference of E
2
and Ell
1
E = (E
2
- E
l )
and must be related to f by the identity
f / +e
0
E
tan = 1-e
0
tan -
(2.4)
The time-of-flight can be written
as Kepler's equation
2
At = E - e sin E
T'
(2.5)
2.3 Transformation of Time-of-Flight to Cubic Equation
The next objective is to rewrite the time-of-flight equation 2.5 in terms
of E and transfer constants. Replace the semimajor axis using the mean point
expression, equation 2.1,
r°P= (1
a
-eo) cos2 2
Then the time-of-flight is
3
1
2i
.At 2(1 - eo) cos2 2E] = E - sin E + (1-e) sin E
srop2
Replace (1 - e) using the identity 2.4 relating f to E,
(1 - e) =
2 tan2 E
2
tan
2
+ tan
2
-
2 2
12
libid., eqn. 7.81.
2
ibid., eqn. 7.80.
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r t
V
t
(t~fan
tan
3
2
tan2
X1 + tan
2
E
2 tan
= E- E+
1 + tan
2
an
2
+
(2.6)
4 tan
3
E
y
tan
2
Define the
dependent
variable for this
method
as
2 E
x tan
2
E
This is convenient
because
ranges from
0 to 2 exclusive
ambiguities. Also define the geometric
parameter
/
tan2f
2
1 - cos f
-1 + cos f
so there
are no sign
(1),2
= i-I
'1+
The
normalized
time-of-flight
equation
is then
(/+ x)(1 + )
x
[( + x)(1+ x) tan' N-x-
( x)]
Figure 3
contains a
graph of
T versus x for various
X. Note
the change
of curvature
in
the X = -0.99 curve.
The simple
Newton's method
will not
work
for
X approaching
-1, or transfer
angles
near 360
°
.
Define
the time-of-flight
parameter,
g.At
2
m
= -
O
P
8gAt
2
-
s3(1 + )6
T
2
- (1 + ,)6
substituting for rop from equation 2.2, so that the time-of-flight equation 2.6
can be written
3
m 2
o)(l + x)
m tan-l
F I
= 2x
_r -+x
ml
(/ + x)(1 + x)
13
+ tan
2
E
T = (1 + )3
(2.7)
+,
(2.8)
-~~~~~~~~~~~~~~~~~~~~~~~~~~~~l
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100
10
T
1
0.1
0.0001 0.001 0.01 0.1 1 10 100 1000 10000
x
Figure 3 Time-of-Flight as a Function of x for Direct Transfer
The equation can be written as a cubic by defining
m
(- + x)(1 + x) (2.9)
All quantities,
l, m, and x, are always positive
so there are no problems in
taking the square root. Then the time-of-flight equation 2.8 is
y3_2_m (ta
+ = 0 (2.10)
The equation can be solved using a successive substitution algorithm:
(1) Make an initial estimate of x.
Inan_1
(2) Calculate
2x
t
-
+x
(3) Solve cubic equation 2.10 for y.
(4) Determine a new value for x by inverting the definition of y, using the
positive sign for the radical since x must be positive.
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(2.11)
(5) Go back to step (2) and repeat until x no longer changes within a specified
tolerance.
A graphic interpretation is shown is Figure 4. An initial value of x is
chosen and used to solve the cubic equation, represented by the vertical line
from the x axis to the thick solid line which is the locus of the positive real
solution to the cubic equation. That value of y is used to solve for a new
value of x, represented by the horizontal line from the locus of the cubic
solution to the thick dotted line which is the locus of the definition of y. The
process is repeated until the solution, the intersection of the solution to the
cubic equation and the definition of y, is reached. Depending on the shape of
the curves, the iteration process may converge quickly or may require many
steps.
cubic
%' definition
x
Figure 4 Solution Path for Direct Transfer*
The examples in this thesis were generated using X= 0 and T = 16.
15
y
1- 2 (M) -
X = 2 + ry2 )
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(2.15)
The successive substitution algorithm is the essentially the same:
(1) Make an initial estimate of x.
(2) Calculate h
1
and h
2
from equations 2.13
and 2.14.
(3) Solve cubic equation 2.15 for y.
(4) Determine a new
value for x from equation 2.11.
(5) Go back
to step (2) and repeat until x no longer changes within a specified
tolerance.
A graphic interpretation of
a successive substitution algorithm is
shown in Figure 5. In the vicinity of the solution the locus of the cubic
solution
is flat, making fewer iterations necessary.
y
'
cubic
I'
definition
·.- ,vv.
I I I I
x
Figure 5 Solution Path
for Direct Transfer Using Free Parameter
17
Y3 ( +li)y2 -h2 = 0
-- ------ 011016 1 -
11 V I
i
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Battin uses a continued fraction
expansion
of tanl
mx
Y;x
singularity at x =0, which is the solution for a parabolic
orbit.
1
2.5 Solution of Cubic Equation
A convenient transformation to use in solving the cubic equation
2.15
y3-(l+hl
) y2 - h2
= 0
is
y = 3 (1 + h
)
+1)
(2.16)
The cubic is converted to the canonical form
z
3
-3z = 2b
if
27h
2
4(1 + hl)3
Let
27h
2
B - 4(1+ h)3
The
solution to the canonical form can
be calculated using the
identities
4s2a-3co 2
4cos a-3cos3' =
cos 2a
4 cosh
3
a - 3 cosh a = cosh2a
3 3
by
libid., pp. 327-328, 334-337.
18
(2.17)
to remove the
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where cosh2a = bfor 1
cosh
2a
= b
f 1
or
2 cos
Z =
cosh
f2osh
fo {1
<
B
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2rlr
2
sin2 2
rl
+ r
2
- 2
rljr
2
cos 2
cos
E
2rlr
2
sin2 (1 + x)
s(l +
)
2
(/
+ x)
The velocity at P
1
required to make the transfer, v
1
, can be written as
1
1
A.
v = ,
1
r
1
+
AX A
lh
x ir,)
A
A
where
irl
is the unit vector in the
direction of r
1
, ih is the unit
vector normal
to the orbital
plane,
and the
quantity a1 is defined
as
1. In
the eccentric anomaly
difference
2
terms of
p and
01 =
0os 0 (
01 % - r c
v
1
= [
cos2 -\
rl
sin
osE) =
s 0 os
sin
rl-x
r
2
r1X)
r
0
r1-x1
2- r2 1+x)
+ sin i
x i
n
2
ih X
r
Substituting
for p
1 2g(1
+x) s(1 +x)
-
rl(-
x) A
= (1+X) s(/+x) rj(1 +x) ir
+
m
n
-h
X
Yr1r 2
The velocity
vL can be
written
in terms of r
1
and
r
2
by
using r
1
and
r2
to
A
calculate i
h
However, this creates a
singularity
if 0 is 1800 at
which point
r,
and r
2
are colinear and cannot
define
a plane.
A rl X r
2
I
h
=
rlr
2
sin 0
1
^
= 1
ir
sin
-
A
x ir
2 )
A
A
h ir
1
A
sin
(ir
2
- cos irl)
20
(2.20)
Then
0
and cos
2
libid.,
eqn.
7.33.
2
ibid.,
p. 307.
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1
-(1
x)
s(l +
,)
2
(1 )
V1
(1
+
)
2s3
)
(2
-
)
r(1
x)
l
2.7
Convergence
Behavior
T
.
8
10
12
14
16
18
20
22
24
26
-0.999
7/8
5/6
5/5
4/5
4/4
3/4
4/4
4/5
4/5
4/5
-0.997
7/7
5/6
5/5
4/5
4/4
3/4
4/4
4/5
4/5
4/5
-0.995
7/7
6/6
5/5
4/5
4/4
3/4
4/4
4/5
4/5
4/5
-0.993
7/7
5/6
5/5
4/5
4/4
3/4
4/4
4/4
4/5
4/5
-0.991
7/7
5/6
5/5
4/5
4/4
3/4
4/4
4/5
4/5
4/5
-0.99
7/7
5/6
5/5
4/5
4/4
3/4
4/4
4/5
4/5
4/5
-0.97
6/7
5/6
5/5
4/5
4/4
4/4
4/4
4/5
4/5
4/4
-0.95
6/6
5/6
5/5
4/5
3/4
4/4
4/4
4/4
4/4
4/4
-0.93
5/6
5/5
4/5
4/4
3/4
4/4
4/4
4/4
4/4
4/5
-0.91
5/6
5/5
4/5
4/4
4/4
4/4
4/4
4/4
4/5
4/5
-0.9 5/6 5/5 4/5 4/4 4/4 4/4 4/4 4/4 4/5 4/5
-0.8
4/5
4/4
3/4
4/4
3/4
4/4
4/4
4/4
4/4
4/5
-0.7
4/4
3/3
3/4
3/4
4/4
4/4
4/5
4/5
4/5
4/5
-0.6
3/4
3/4
4/4
4/5
4/5
4/5
4/5
4/5
5/5
5/5
-0.5
3/4
4/4
4/5
4/5
4/5
5/5
5/5
5/5
5/5
5/5
-0.4
4/5
4/5
4/5
4/5
5/5
5/5
5/5
5/5
5/5
5/6
-0.3
4/5
5/5
5/5
5/5
5/5
5/5
5/5
5/6
5/6
5/6
-0.2
4/5
5/5
5/5
5/5
5/6
5/6
5/6
5/6
5/6
5/6
-0.1
5/5
5/5
5/5
5/6
5/6
5/6
5/6
5/6
5/6
5/6
0
5/5
5/5
5/6
5/6
5/6
5/6
5/6
5/6
5/6
5/6
0.1
5/5
5/5
5/6
5/6
5/6
5/6
5/6
5/6
5/6
5/6
0.2
5/5
5/5
5/6
5/6
5/6
5/6
5/6
5/6
5/6
5/6
0.3
5/5
5/5
5/6
5/6
5/6
5/6
5/6
5/6
5/6
5/6
0.4 5/5 5/5 5/6 5/6 5/6 5/6 5/6 5/6 5/6 5/6
0.5
5/5
5/5
5/6
5/6
5/6
5/6
5/6
5/6
5/6
5/6
0.6
5/5
5/5
5/5
5/6
5/6
5/6
5/6
5/6
5/6
5/6
0.7
5/5
5/5
5/5
5/6
5/6
5/6
5/6
5/6
5/6
5/6
0.8
5/5
5/5
5/5
5/5
5/6
5/6
5/6
5/6
5/6
5/6
0.9
5/5
5/5
5/5
5/5
5/5
5/6
5/6
5/6
5/6
5/6
0.91 5/5
5/5
5/5
5/5
5/5 5/6
5/6
5/6 5/6
5/6
0.93
5/5
5/5
5/5
5/5
5/5
5/5
5/6
5/6
5/6
5/6
0.95
4/5
5/5
5/5
5/5
5/5
5/5
5/6
5/6
5/6
5/6
0.97
4/5
5 5
5/5
5/5
5/5
5/5
5/6
5/6
5/6
5/6
0.99
4/5
5/5
5/5
5/5
5/5
5/5
5/5
5/6
5/6
5/6
0.991
4/5
5/5
5/5
5/5
5/5
5/5
5/5
5/6
5/6
5/6
0.993
4/5
5/5
5/5
5/5
5/5
5/5
5/5
5/6
5/6
5/6
0.995 4/5 5/5 5/5 5/5 5/5 5/5 5/5 5/6 5/6 5/6
0.997
4/5
5/5
5/5
5/5
5/5
5/5
5/5
5/6
5/6
5/6
0.999
4/5
5/5
5/5
5/5
5/5
5/5
5/5
5/6
5/6
5/6
Table
1
Number
of Iterations
to Compute
Direct
Transfer
Solution
to
Eight/Twelve
Significant
Figures
21
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Table 1 contains the number of iterations required for the direct
transfer algorithm to converge to at least eight and twelve significant figures
for various T and X. Note that only one additional iteration is required to
solve for the additional four significant figures. The initial estimate of x used
was / as suggested by Battin and Vaughan.
1
The only difficult problem seems
to be for transfers near 360
°
, X = -1, for smaller T that approach the minimum
energy period of 2.2
p. 47.
1
Vaughan, Robin M. An
Improvement of Gauss' Method for Solving Lambert's Problem,
2
ibid., Table 8.
22
- -
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Chapter
3 Time-of-Flight Equation
with Multiple Revolution
Orbits
3.1 Addition
of Multiple Periods
Because the orbital period
is just a function of the semimajor axis,
P=2
a
3
/ g, Lambert's theorem
can be extended
to include multiple
revolutions by
the inclusion of the number, N, of complete
revolutions in
the orbital transfer,
4At
= F (N, a, r
l
+ r
2
, c)
and the geometrical
orbit transformation
for the direct
transfer case remains
valid.
Kepler's
equation for
the multiple revolution orbit must
be
At = (N + E) - e sin E
Writing
a and e in terms
of rp, E, and f as in section 2.3 yields
4 tan
3
, , ,, ,, ,, ,,,, , ,,
8rpAt
[(tan2
+ tan
2
1
(3.1)
3
+
tan
2
)I 2
E
2 tan 2
= (N +E)-
E2
1 +
tan
2
2
4 tan
3
E
2
2
+
tan
2
1
+ tan
2
E
Using
the same
definitions as for
the direct
transfer
T = (1 + )3
( + x)(1 + X)
x
- ( - x)] (3.2)
which, of course, reduces
to the direct
transfer case, equation 2.7,
if N = 0.
From Figure 6 of T
versus x for N = 0
and N = 1 note that there are two
solutions
given T, X, and N
greater than
0 for T > Tm and one
solution
for
23
-~~~~~~~~~~~I
R/ +x)(1 + x
(Nc+
tan-1 -X
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T = Tm, where
Tm is the minimum of the T versus x curve.
It would appear
that solutions for T near Tm will be difficult to obtain for X near +1 because the
locus of T is flat for a wide range of x.
100
10
T
1
0.1
0.0001
Figure 6
0.001 0.01 0.1 1 10 100
x
1000 10000
Time-of-Flight
as a Function of x for Multiple Revolution
Orbits
3.2 Estimation of Maximum Number of Revolutions
To know if a
solution will be possible within the time
constraint,it is
necessary to be able to determine Tm given
X
and N. Differentiate the time-of-
flight equation 3.2,
dT (1 +X)
3
x
dx-
4x
3
\1(l
+
x)(l
+
x)
x{I [3x+ 23/)x
+ (3 +
2/)x + 3/ 2
-3(-x2)(/+
)( + x)(N
2
+
and set equal to
zero to solve for xm where T
(xm) = Tm,
-
[3x
3
+ (2
+ 3/)x
2
+
/
(3 + 2/)xm
+ 3/ 2]
-3(/x)(/ +Xm)(1xm)(Nan' ) =0
24
tan'
-1
)
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N=1
Xm
X approx xm
1
N=10
Im Xm
L:"M approx x
-1
0
1
X
Figure 7 xm and Approximation as a Function of X
This certainly may be inaccurate if the
right-hand expression is close to an
integer, but it gives an approximation of Nmax.
26
Ii
I~~~~~~~~~~~~~~~~~~~
L =
= I I I=
=K~~~ I= I =
~ ~a
i-1-" .........d
20
x
0
-1
40
x
0
0
X
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N=1
-1 0
1
1%
N=10
T
m
T (approx x
)
- approx T
-1 0 1
Figure 8 Tm and Approximations as a Function of
X
In sum there are
2
Nmax
or (
2
Nmax+l) possible solutions
to a given
problem. For every N from 1 to (Nmax-1) there are two solutions. There is
one solution
for the direct transfer,
N = 0, case, and there may be one or two
solutions
for N = Nmax, depending on whether
T is greater than or
equal to
Tm. Obviously
the transfer orbit with
the largest semimajor axis (and energy)
results from
a direct transfer and the orbit with the smallest
semimajor axis
27
13
T
A;
70
T
I
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ly ?COPlete
?eoll,,,
Oi°
ible
-y
is
graphof a
iSC"iS , -n
79 9
1
28
t(i iSaC
thet
P-y
-4..V.Is
I)
l
4:, f7
- I. , -16
4>' -1
' -I,
I
m
l lj n
-PighsY- -
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Chapter 4 Algorithm to Determine
Low EnergySolution
4.1 Revised Cubic
Equation
The time-of-flight equation
3.1 is
[(m
2 InN2] + tan
1
x
1 x)( x
(/
+
)( + x) 2x + )(1 +
As before let
2
m
Y2 x +x
t + x)(l
+x)
Then the cubic equation
is
tan
'
,~
m(
3 - y2 2x
+x=
-
cubic
-' definition
x
Solution
Path for Multiple
Revolutions*
The multiple revolution examples were generated using X= OT = 16, and N = 1.
29
(4.1)
Y
Figure 10
m
-
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which reduces
to the direct transfer equation 2.10 when N = 0.
Figure 10 shows the solution of the cubic equation 4.1 and the
definition of y, equation 2.9, versus x. Note that the solution of the cubic goes
to infinity as x approaches zero. Also the inversion of the definition of y
yields only one positive x. This makes it impossible to use successive
substitution to get the lower x solution. This algorithm can only yield the
higher x solution. The shape of the curves may make the successive
substitution solution more difficult to obtain than for the direct transfer.
4.2 Choice of Free Parameter to Flatten Cubic Equation
To flatten the cubic solution in the vicinity of a root introduce the free
parameter hi as in section 2.4,
3- _ (han-2_
)
anhi
~y
2x
_x T~~ /
x(1 +)
(42)
Differentiate
[3y
2
- 2(1 + hi
)
] [Ym dh
1
dy
s
At the solution point the second term is zero. Choose hi so that is zero.
N1 + tan il
d-x I+x h1 x (/+ x)(1+ x) =0
(/ )2 N an
1
F1
i = 23 (1 +x(3 + 5x) (4.3)
4x2( + 2x + ()
~x
30
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hi
1
( + x)(1 +x)J
m {r
4x
2
(1 + 2x +
/)
[
x
2
(1 +
/)x
-
36/
N2+ tan-
1
-x
I2 + (3/ + x)
N-~
(4.4)
As expected,
the equations reduce
to those
of the direct transfer
case,
equations 2.13 and 2.14.
The cubic
equation 4.2 is of
the form
y3-(l+hl
) y 2 - h 2
= 0
(4.5)
A graphic interpretation
of a successive
substitution algorithm
using
this
equation is shown
in Figure 11. The
solution of the
cubic equation
4.5 is
shown
as zero if the
cubic does not have
a real
positive solution.
y
cubic
MM definition
x
Figure 11
Solution Path for
Multiple
Revolutions Using
Free Parameter
31
Let
an'
h2 m 2x t -
+x)
+
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Note the "dead" area where there is no positive real solution
to the cubic
equation. Near x = 0 the root of the cubic is very large,
eliminating the low x
solution completely from the algorithm. An initial
estimate of x that is large
enough
to avoid this "dead" area is needed. The
locus of the solution of the
cubic is very flat, similar to the direct transfer, requiring fewer iterations of
the successive substitution algorithm to reach the solution.
32
-
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try
5.1
edefl1t3°'
In
order
cubic
eq
u a
fio
y
(
-I +-)
1 -
(I
+ -A
rewrite
the
tire
oott
ath
Otion
s'x: '-e
:oot of the
5.1)
Then
N\ (5.2)
1 +)
Pdefiea
Y
33
sol-atito,
,,.~et
Agpilthvh
t.
etevdnine
,-U
'Ote'r
-
.-~~~~~~
vert J
1ca)(
~tlkt
Qua
on 3
A
3
ra~
-1
+tanlr
[
- 2
x
..--4
-
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Using the new definition of y, equation 5.1, x can be calculated as
(5.3)
Figure 12 shows new y versus x curves. Both solutions are possible
using successive substitution by using the "+" to get the higher x estimate and
"-" to get the lower x estimate. Of greatest importance, however, is that the
cubic solution curve appears to be "flattenable" in the vicinity of the lower x
solution.
5.2 Choice of Free Parameter to Flatten Cubic Equation
Following same procedure as before, introduce a free parameter h
1
,
y
3
- (1 + h
)
y
2
-
+x
(
hl
+ )(1 +x) 0
Differentiate
[3y
2
- 2 (1 + h
1
) y]
_j y2
- mx{df1 N+ an
-mX~l;x -TX
W
_
m
1
dhl
(/+ x)(1 + x) dx -
1 -hl d
+x Jhlx
(1 + h
1
) y
2
1
+
+ )(1 + X)j
h dxW
x)(l+x) dx =0
Use the definition of y, equation 5.1, to
dy
eliminate terms and set a to zero
34
(5.4)
tan
-1
-
1 +x
( +
X m + I
2 - (l+/(1 ;7-(l+ .( -V
2x + tan-1 1Jx
- x~x1 2
1 T+
- Tx
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mF3
N2+
an
1
2 2x
\F
1 A (2h
t
- 1)
1 +x) + x)(1+ x)
d
+
l
42+tani-
1
-m
-x
1 +
x)
1
1
=0
Carrying
through the algebra,
the result
is a
very simple neat
formula
for hi,
h
1
=
(
+x)(1+
2x+/)
(5.5)
Obviously there is
a singularity
at
'/7. However, since
the
smaller x solution
must be
less than
f7/, the singularity
is
outside
the range of interest
of x.
Solve
for h
2
1
h
1
++
x)(1
+x)
x2)+
tan
-
1
x
_ X2) 2 _
(r + x)
h2= 2
- x2)
/
(5.6)
As
desired,
the cubic equation
has a
finite solution
at zero.
Figure
13 displays
the
y versus x curves
using
the free
parameter to
flatten
the cubic.
Again for
ranges
of x for which
there is
no positive
real
solution
of the cubic,
the cubic
solution
is shown
as
zero. The
above
definition
of h
1
eliminates
the possibility
of
using this
definition of y
to find
both solutions.
In the
vicinity
of the lower
x solution,
the
locus of the
cubic
solution
is
very flat, requiring
very
few iterations
for solution
if
a small
enough initial estimate of x is used. Near \F there are positive real roots to
the cubic
equation, but
they are
very large
and irrelevant.
35
-h-
TV+X;(+ )f
h,[2xI~~~I
an-i~F
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-- cubic
- definition
x
Solution Path for Multiple Revolutions
Using Redefined y and
Free Parameter
5.3 Solution
of Cubic Equation
The cubic
equation is now of the form
y
3
- (1+
)
y
2
-h
2
= 0
Use the transformation
= ',3x(1
+
h)
(b
+ 1)
(5.8)
but
note that it introduces a singularity at zero.
The cubic is converted to the
canonical form
z
3
-3z
= 2b
if
27h
2
4[LJ-(1
+ hl)]
3
1
36
y
Figure
13
(5.7)
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Define
B -
27h
2
4[
`x(1 + hl)]
3
From
section 2.5,
for {
2 cos[3
2 cosh
For
the high
energy
solution
B is almost
alwa
quite
large, except
for T near
Tm.
For B> 0 consider
(5.9)
B < 01
B
positive
and is often
ys positive and
is often
2
z = 2 cosh a where cosh 2a = b
Using
identities for
hyperbolic
functions
write
2a =
cosh
- l
b =
ln (b + b
2 7
T
-
1
)
1
z
=
2cosh[
'ln(b
-1i)]
= eln(b
+ bi
1)3+e-ln
(b+
~~~~~~~
1
1
z = (B+
B+1)3+('J+
B+) 3
The root
of the
high-energy
cubic
is
where
1
B+ 1
)3
37
1
N/b
2
_ )5'
2
B+
+
I_ +
y
=
T+1/A
2
~- -JB+ TA + 11
= ( + h,[
A2
2+1
1(510)
-
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..
A~~'
01,&
V
XOTu~O
.- xe'%g~
A
.0
..__
k~ao
ea,
Cotnposit.,~r,,hosen SO~ ,
x -- ,
soX01LO0
v~.Aa vitw
'h
) C7
type °M~ 1~t15
...
t,,eVN -
16z_
cx'g
Cf%,T
V--
,,the
sv~O
,Vas
)L~
opV Ah9
'a,c toO
If
-
8/16/2019 An Elegant Lambert Algorithm for Multiple Revolution Orbits - Loechler
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(
+ x)2
h1
=
4x2(1 2x+ )
N + tan
'1
3(1 +x)
2
r -(3 +
N
+
tan-
1
\/*
+ (3/
+ x)}
so
(1 + h) =
1
4x
2
(1 + 2x
+,')
{3(x)
2
(1 +
x)
2
N' + tan
-
1
-f
+ [3x
3
+ (1-6/)x
2
-/6+5/)x - 3/2
The easiest
way to guarantee that B >-1
is to require
that (1 + hi) and
h
2
are
both positive.
h
2
will be definitely positive
if
x
2
-
(1 + )x - 3/
0
x(1+
+i
12/
+ 1+/)2
Use the binomial
series expansion
of the
square root
12 1 12 2
2 (1+1~~~2 1'(
3/
1+"2
1 +/
xo=1
+4/
1
+4/
(6.1)
Now check that this value
will yield a positive
(1 + hi)
by substituting into
the only term that
could be negative
3(1+ 4d)3 (1 - 6/)l + 4)2-/ (6 + Z)(1+ 4) _ 3/ 2
4+32 + 802+76/3 > 0
Therefore x
= 1 + 4/ is an
acceptable initial
estimate for the low
energy
algorithm.
39
h = 4x
2
(1
5x)
m
+
x+[x
2
- ( +x - 3
X >
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For the
high energy algorithm
hi will be positive for all x less than
but determining at what
x h
2
would be positive is difficult. Instead consider
that at x = 0, which is definitely
smaller than the solution, y
3
equals h
2
.
1
y (x=O) = (mN)
3
From equation
5.3 find the next value
of x,
1 1
XO
= 2m(O +3- - 32 (-4/ (6.2)
N2
m
N2
6.2
Summary of Algorithms
The multiple revolution
algorithms can be summarized,
with the
appropriate
equation numbers, as follows:
Low Energy
High Energy
(1) Make an initial estimate of x.
6.1 6.2
(2) Calculate hi and h
2
.
4.3 and 4.4 5.5 and
5.6
(3) Calculate
B.
2.17 5.9
(4) Calculate
z. 2.18
2.18
(5) Calculate y.
2.16
2.16 or 5.10
(6) Determine a new value for x.
2.11 5.3
(7) Go back to step
(2) and repeat until x
no longer changes
within a specified
tolerance.
6.3 Convergence Characteristics
Tables 2 and 3 display the
number of iterations
required to solve for
the
low energy solution
to at least eight
and twelve significant figures for N = 1
and N = 2
respectively. As with
the direct transfer algorithm, only one
additional iteration is required to find the
additional four significant
figures.
In general,
solution is more difficult near Tm, but
especially so if X is near
±1.
40
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T
X 8 10 12 14
16 18 20 22 24
26
-0.999
34/35 9/10 7/7
6/6 5/6 5/5
4/5 5/5
-0.997
28/28 9/9
7/7 6/6
5/6 5/5
4/5
5/5
-0.995 24/25 8/9 7/7
6/6 5/6 5/5
4/5 5/5
-0.993 21/22 8/9 7/7
6/6 5/6 5/5 4/5 5/5
-0.991
19/20 8/9 7/7 6/6 5/6
5/5 4/5 5/5
-0.99
18/19 8/9 7/7 6/6
5/6 5/5 4/5
5/5
-0.97 11/12 7/8 6/7
5/6 5/5 4/4
5/5 5/5
-0.95 9/10
7/7 6/6 5/6
5/5
4/5 5/5 5/5
-0.93
8/9 6/7 6/6 5/6 4/4 5/5
5/5 5/5
-0.91
8/8
6/7 5/6 5/5 4/5 5/5
5/5 5/5
-0.9
7/8 6/7 5/6 4/5 5/5 5/5
5/5 5/5
-0.8
6/6 5/5 5/5 5/5 5/5
5/5 5/5 5/5
-0.7
7/8 5/5 5/5 5/5 5/5
5/5 5/5
5/5 5/5
-0.6 6/7 5/5
5/5 5/5
5/5 5/5 4/5 4/5
4/5
-0.5
5/6 5/6 5/6
5/5 5/5
4/5 4/5
4/5 4/4
-0.4
5/6 5/5
5/5 5/5 5/5
4/5 4/5 4/4 4/5
-0.3
5/6 5/5 5/5 5/5 4/5
4/5 4/4
4/5 4/5
-0.2 5/6
5/5 5/5 4/5 4/5 4/4 4/4
4/5 4/5
-0.1
5/6 4/5
4/5
4/5
4/5 4/4 4/4
4/4
0
5/6
4/5 4/5 4/5 4/4
4/4 4/4
4/4 4/4
0.1 5/6
4/5 4/4 4/4 4/4 3/4 3/4
3/4 3/4
0.2
6/6 4/5 4/5 4/4 3/4 3/4
3/4 3/4 3/4
0.3
6/6 5/5 4/5 4/4 3/4
3/3 3/3
3/4 3/4
0.4 6/6 5/5
4/5 4/4
3/4 3/4 3/3 3/3
3/4
0.5 6/6 5/5
4/5 4/4 4/4
3/4 3/4 3/4 3/4
0.6
6/6 5/5
4/5 4/5 4/4 3/4
3/4 3/4
3/4
0.7 6/6 5/6 5/5
4/5 4/4 4/4
3/4 3/4 3/4
0.8
6/6 5/6 5/5
4/5 4/5 4/4 4/4 3/4 3/4
0.9
8/9 6/7 5/6
5/5 5/5 4/5
4/5 4/4 4/4 3/4
0.91
8/9 6/7
5/6 5/6 5/5
4/5
4/5 4/5
4/4 4/4
0.93
8/9
6/7 5/6 5/6
5/5 4/5
4/5 4/5 4/4
4/4
0.95 8/9 6/7
5/6 5/6
5/5 4/5 4/5
4/5 4/4
4/4
0.97 8/9 6/7 6/6 5/6 5/5
5/5 4/5 4/5
4/4 4/4
0.99
8/9
6/7 6/6 5/6
5/5 5/5
4/5
4/5
4/5
4/4
0.991
8/9 6/7 6/6 5/6
5/5 5/5
4/5 4/5
4/5 4/4
0.993 8/9 6/7 6/6
5/6 5/5
5/5 4/5
4/5 4/5 4/4
0.995 8/9 6/7
6/6 5/6
5/5 5/5
4/5 4/5 4/5 4/4
0.997 9/9
6/7
6/6 5/6
5/5 5/5 4/5
4/5
4/5 4/4
0.999 9/9
6/7 6/6 5/6 5/5
5/5 4/5
4/5 4/5 4/4
Table 2
Number of Iterations
to Compute Low Energy
Solution for
N = 1
to
Eight/Twelve
Significant
Figures
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42/48
T
8 10 12 14 16 18 20 22 24 26
-0.999 11/12 8/9 7/7 6/7
-0.997
11/11 8/8 7/7 6/7
-0.995 10/11 8/8 7/7 6/7
-0.993 10/11 8/8 7/7 6/7
-0.991
10/10
8/8 7/7 6/7
-0.99
10/10
8/8 7/7 6/7
-0.97
14/15 8/9 7/8 6/7
6/6
-0.95
11/11 8/8 7/7 6/7 6/6
-0.93
9/10 7/8 6/7 6/6 5/6
-0.91
9/9 7/8 6/7 5/6
5/6
-0.9 8/9 7/7 6/7 5/6 5/6
-0.8
7/7
6/6 5/6 4/5
5/5
-0.7
6/7 5/6
4/5 5/5 5/5
-0.6
8/9
5/6 4/5
5/5
5/5 5/5
-0.5
7/8
5/6
4/5 5/5 5/5
5/5
-0.4
7/7
5/6
4/5
5/5 5/5
5/5
-0.3 6/7 5/6 4/5 4/5 5/5 4/5
-0.2
6/7
5/6
4/5
4/5 4/5
4/5
-0.1
6/7 5/6
4/5
4/5 4/5
4/5
0
6/7
5/6
5/5 4/5
4/4
3/4
0.1
7/7
5/6 5/5
4/5
4/5 4/4
0.2
7/7 5/6
5/5
4/5 4/5
4/4
0.3
7/7
5/6 5/5
4/5
4/5 4/4
0.4
7/7
5/6 5/5
4/5
4/5 4/4
0.5
7/7
5/6
5/5
5/5 4/5
4/5
0.6
7/7
5/6
5/6 5/5
4/5
4/5
0.7
7/7
6/6
5/6 5/5
5/5
4/5
0.8
7/7
6/6 5/6
5/6
5/5
5/5
0.9
7/8
6/7
6/6 5/6
5/6
5/5
0.91 7/8 6/7 6/6 5/6 5/6 5/5
0.93
11/12 7/8
6/7
6/6 5/6
5/6
5/5
0.95
10/11
7/8 6/7
6/6
5/6
5/6
5/6
0.97
10/11
8/8 6/7
6/7
6/6
5/6 5/6
0.99
11/11
8/8
7/7 6/7
6/6
5/6
5/6
0.991
11/11
8/8 7/7
6/7
6/6
5/6
5/6
0.993
11/11
8/8
7/7 6/7
6/6
5/6 5/6
0.995
11/12
8/9
7/7 6/7
6/6
5/6
5/6
0.997
11/12
8/9 7/7
6/7
6/6
5/6
5/6
0.999
11/12
8/9
7/7
6/7 6/6
5/6
5/6
0i999
Table 3
Number
of Iterations
to Compute
Low Energy Solution
for N
= 2
to Eight/Twelve Significant Figures
Tables 4 and
5 display the number
of iterations required
to solve for the
high energy solution
to at least eight
and twelve significant
figures for N = 1
and N
= 2 respectively.
As
with the direct transfer
algorithm
and the low
energy
algorithm, only one
additional
iteration
is required to find
the
42
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additional
four significant
figures.
Unlike the low
energy
algorithm,
solution
is not
more
difficult
for
T near
Tm or
for
near+1.
T
X
8
10
12
14
16 18
20 22
24
26
-0.999
6/6 4/5
4/4
4/5
4/4 4/4
4/4 4/4
-0.997
5/6 4/* 4/5
4/4
4/5 4/4
4/4 4/4
-0.995
5/6
4/5
4/5 4/4
4/4
4/4
4/4 4/4
-0.993
5/6 4/5
4/5
4/4
4/4
4/4
4/4
4/4
-0.991
5/6
4/5 4/5
4/4
4/4
4/4 4/4
4/4
-0.99
5/6
4/5
4/5
4/4 4/4
4/* 4/4
4/4
-0.97
5/6
4/5 4/4
4/4
4/4 4/4
4/4
3/4
-0.95
5/5
4/5
4/4 4/4
4/4
4/4 4/4
3/4
-0.93
5/5
4/5
4/4
4/4 4/4
4/4
4/4 3/4
-0.91
5/5
4/5 4/4
4/4
4/4 4/4
3/4
3/4
-0.9
4/5
4/5 4/4
4/4
4/4 4/4
3/4
3/4
-0.8
4/5
4/4 4/4
4/4
4/4
3/4 3/4
3/4
-0.7
5/6 4/5
4/4
4/4 4/4
3/4
3/4
3/4 3/4
-0.6
5/5
4/4
4/4 4/4
3/4
3/4 3/4
3/4 3/4
-0.5
5/5 4/5
4/4
4/4 3/4
3/4
3/4
3/4
3/4
-0.4
4/5
4/4
4/4
4/4
3/4 3/4
3/4
3/4 3/4
-0.3
5/5
4/4
4/4 3/4
3/4
3/4
3/4
3/4
3/4
-0.2
5/5
4/4
4/4
3/4
3/4
3/4 3/4
3/4
3/4
-0.1
4/5
4/4
4/4
3/4 3/4
3/4
3/4 3/4
3/4
0
4/5
4/4
4/4 3/4
3/4 3/4
3/4
3/4 3/4
0.1
4/5
4/4
4/4
3/4
3/4
3/4
3/4
3/4
3/4
0.2
4/5
4/4 4/4
3/4
3/4 3/4
3/4 3/4
3/4
0.3
4/5
4/4
4/4
3/4 3/4
3/4
3/4
3/4
3/4
0.4
4/5 4/4
4/4
4/4
3/4
3/4
3/4 3/4
3/4
0.5
4/5 4/4
4/4 3/4
3/4
3/4 3/4
3/4 3/4
0.6
4/5
4/4
4/4
3/4
3/4
3/4
3/4
3/4 3/4
0.7 4/5 4/4 3/4 3/4 3/4 3/4 3/4 3/4 3/4
0.8
4/4
3/4
3/4 3/4
3/4
3/4 3/4
3/4
3/3
0.9
5/5
4/4 3/4
3/4
3/4
3/3
3/3
3/3 3/3
3/3
0.91 4/5
4/4
3/4
3/4 3/4
3/3
3/3 3/3
3/3
3/3
0.93
4/5 3/4
3/4
3/4
3/3
3/3 3/3
3/3
3/3 3/3
0.95
4/5
3/4
3/4
3/4 3/3
3/3
3/3 3/3
3/3
3/3
0.97
4/4
3/4 3/3
3/3
3/3 3/3
3/3
3/3
3/3 3/3
0.99 3/4
3/3
3/3 3/3
3/3
2/3
2/3 2/3
2/3
2/4
0.991
3/4
3/3 3/3
3/3
2/3
2/3
2/3 2/4
2/3
2/3
0.993
3/4
3/3
3/3 2/3
2/3
2/3
2/3
2/3 2/3
2/3
0.995
3/3
3/3
2/3
3/3
2/3
2/3
2/3
2/3
2/4 2/3
0.997
3/3
3/4 2/3
2/3
2/3
2/3
2/3 2/3
2/3
2/3
0.999
3/3
2/3
2/3 2/4
2/2
2/2
2/2
2/2 2/2
2/3
Tab
ile
4
Number
of Iterations
to Compute
High Energy
Solution
for
N
= 1
to Eight/Twelve
Significant
Figures*
*Does
not
converge to
twelve
significant
figures
with
current implementation.
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T
8
10
12
14
16
18
20
22
24
26
-0.999
5/5
4/6
4/5
4/4
-0.997
5/5
4/5
4/4
4/4
-0.995
5/5
4/5
4/4
4/4
-0.993
5/5
4/5
4/4
4/4
-0.991
4/5 4/5 4/4 4/4
-0.99
4/5 4/5
4/4
4/4
-0.97
5/6 4/5
4/5
4/4
4/4
-0.95
5/6
4/5
4/4
4/4
4/4
-0.93
5/5
4/5
4/4
4/4
4/4
-0.91
5/5 4/5
4/4
4/4
4/4
-0.9
5/5
4/5
4/4
4/4
4/4
-0.8
4/5
4/4
4/4
4/4
3/4
-0.7
4/5
4/4
4/4 3/4
4/4
-0.6
5/6
4/5
4/4 3/4
4/4
3/4
-0.5
5/6 4/4
4/4
4/4
3/4
3/4
-0.4
5/5
4/5 4/4
4/4 3/4
3/4
-0.3
5/5
4/5
4/4 4/4
3/4
3/4
-0.2
5/5
4/5
4/4 4/4 3/4 3/4
-0.1
5/5
4/4
4/4
4/4
3/4
3/4
0
5/5
4/4
4/4
3/4
3/4
3/4
0.1
5/5
4/4
4/4
3/4
3/4
3/4
0.2
5/5
4/4
4/4
3/4
3/4
3/4
0.3
5/5
4/4
4/4
3/4
3/4
3/4
0.4
5/5
4/4
4/4
3/4
3/4
3/4
0.5
4/5
4/4
4/4
4/4
3/4
3/4
0.6
4/5 4/4
4/4 3/4
3/4
3/4
0.7
4/5 4/4
4/4
3/4 3/4
3/4
0.8
4/5
4/4
3/4
3/4
3/4
3/4
0.9
4/4
3/4
3/4
3/4
3/4
3/3
0.91
4/4 3/4
3/4
3/4 3/4
3/3
0.93
5/6 4/4
3/4 3/4 3/4 3/3 3/3
0.95
4/5
3/4 3/4
3/4
3/3
3/3
3/3
0.97
4/4
3/4
3/4
3/3
3/3
3/3
3/3
0.99
3/4 3/3
3/3 3/3
3/3
3/3 3/3
0.991
3/4
3/3
3/3
3/3
3/3
3/3
2/3
0.993
3/4
3/3
3/3
3/3
3/3
2/3
2/3
0.995
3/4 3/3
3/3
2/3
2/3
2/3 2/3
0.997
3/3
3/3
3/3
2/3 2/3
2/4
2/3
0.999
3/3 2/4
2/3
2/3
2/3
2/3
2/2
Table
5
Number
of
Iterations
to
Compute
High
Energy
Solution
for
N
= 2 to
Eight/Twelve
Significant
Figures
Table
6 displays
the
number
of
iterations
required
to
compute
both
solutions
to
at least
eight
and
twelve
significant
figures
for
T = 1.OlTm.
The
property
that
only
one
additional
iteration
is needed
to compute
the
additional
four
significant
figures
is maintained.
There
are
no difficulties
in
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finding
the high energy
solution,
but many
iterations are
required
to find
the
low energy solution
near
the singular
points at X= ±1.
X
Tm
Low Energy
High Energy
-0.999 11.63781258943 33/34 6/7
-0.997
11.60361802781
28/29
6/7
-0.995
11.57018940617
26/27
6/7
-0.993 11.53751862029
24/25
6/7
-0.991
11.50559482845 23/24
6/7
-0.99 11.48990898153
23/23
6/7
-0.97
11.21121489822
16/17
6/7
-0.95 10.98572795637
14/14 6/7
-0.93 10.79726396256
12/13 6/7
-0.91
10.63549866068 11/12
6/7
-0.9 10.56251463024
11/12
6/7
-0.8
10.02008404139
9/10
6/7
-0.7
9.68146547180
8/9
6/6
-0.6 9.45927663312 8/8 6/6
-0.5
9.31413909263
7/8
6/6
-0.4
9.22304335083
7/8
6/6
-0.3 9.17032549577
7/8
6/6
-0.2
9.14412122311
7/8
6/6
-0.1
9.13466385734
7/8
6/6
0
9.13332658859 7/8
6/6
0.1
9.13198931985
7/8
6/6
0.2
9.12253195403
7/8
6/6
0.3
9.09632767791 8/8
6/6
0.4
9.04360975307
8/8 6/6
0.5 8.95251322580
8/8
6/6
0.6
8.80736926187
8/9
6/6
0.7 8.58513508118 8/9 6/6
0.8
8.24619104536
9/10
6/6
0.9 7.70058452852
10/11
5/6
0.91
7.62652569540
10/11
5/6
0.93
7.46118463150
11/11
5/6
0.95 7.26508215591
11/12 5/6
0.97
7.02000399780
13/13
5/6
0.99
6.66866780554
16/17
5/5
0.991
6.64486144792
16/17
5/5
0.993
6.59356093535
17/18
5/5
0.995 6.53561938625
18/19
4/5
0.997
6.46700406156
21/21
4/5
0.999 6.37505540838
25/26
4/4
Number
of Iterations to
Compute
Solutions for N
= 1 at 101%
of
Tm to Eight/Twelve
Significant
Figures
45
Table 6
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46/48
Chapter 7 Conclusion
Multiple revolution transfer orbits are considered as solutions to
Lambert's
problem by adding the number of complete revolutions to Kepler's
equation.
This creates a double-valued time-of-flight
with a clearly defined
minimum which permits the solutions to be distinguished by their
relative
energy. Therefore two solutions exist for each complete
revolution if the
given transfer time is greater
than the minimum time-of-flight; one solution
exists if the given transfer time is equal to the minimum
time-of-flight; and
no solutions
exist if the given transfer time is less than
the minimum time-
of-flight. In general there are (
2
Nmax+l) possible solutions
to a given problem
with the direct transfer orbit having the most
possible energy or largest
semimajor axis and the lower energy, smaller semimajor axis, solution for
the Nmax revolution orbit having the least
possible energy.
The multiple revolution algorithms
are based on Battin's direct trans-
fer algorithm which transforms the mean point, the point at which
the or-
bital tangent is parallel to the chord connecting the two terminal points,
to a
1
pericenter or apocenter and which uses as a dependent variable x - (E
2
- El).
A variable y is defined so that Kepler's equation
can be formulated as a cubic
equation in y. A successive substitution method is used
by estimating x,
solving the cubic
equation for y, and calculating the next estimate of x from
the definition
of y. The method is quick and efficient
because of the addition
of a free parameter to the cubic equation
which is chosen to make the slope
zero at the solution
point. This guides the estimation
of x to the solution
point, requiring few iterations to converge.
The low energy algorithm is derived using the same definitions as
for
the direct transfer
solution and, in fact, reduces to
the direct transfer
algorithm if zero is used as the number of complete
revolutions. The low
energy algorithm
can only converge to the low energy solution. The high
energy
algorithm is derived using a definition
of y that makes the cubic
solution finite for small x. The choice of the free parameter makes
the high
energy algorithm converge
to only the high energy solution.
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The
convergenceof both
the low
and high
energy algorithms
compares
favorably
with that
of the direct
transfer. The
characteristic
of
improving four
significant
figures
with
one iteration
once the
solution
point has been
reached
is retained.
Less than
ten iterations,
and
often only five,
are required
for almost
all
problems.
Convergence
is nearly
completely
uniform for
the
higher
energy algorithm
while
the lower energy
algorithm
requires
a large
number of iterations
only if the
given transfer
time is slightly
greater than
the
minimum
time-of-flight
and the transfer
angle
is either
slightly larger than
0
°
or slightly
smaller
than 360
°
.
Computational
issues
must still be
considered,
but the multiple
revolution
algorithms'
convergence
behaviors
indicate
great capability
for
handling a
wide variety
of navigational
problems.
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References
BATTIN, RICHARD H.
An Introduction
to the Mathematics and Methods of
Astrodynamics,
American
Institute of Aeronautics and Astronautics,
Inc., 1987.
BATTIN,
RICHARDH. and VAUGHAN, ROBIN
M. "An Elegant Lambert
Algorithm," Journal of Guidance, Control, and Dynamics,
Vol. 7, No. 6,
Nov.-Dec. 1984, pp. 662-670.
SUN, FANG-TOH, VINH, NGUYEN
X., and CHERN, TZE-JANG.
"Analytic Study
of
the Solution Families
of the Extended
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